metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D10, (C22×C4)⋊3D5, (C22×C20)⋊2C2, C10.42(C2×D4), (C2×C10).37D4, (C2×C4).65D10, C23.D5⋊6C2, D10⋊C4⋊2C2, C10.D4⋊3C2, C10.18(C4○D4), C2.18(C4○D20), (C2×C10).47C23, (C2×C20).78C22, C5⋊4(C22.D4), C22.9(C5⋊D4), (C22×D5).9C22, C22.55(C22×D5), (C22×C10).39C22, (C2×Dic5).15C22, C2.6(C2×C5⋊D4), (C2×C5⋊D4).6C2, SmallGroup(160,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd9 >
Subgroups: 240 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.23D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C5⋊D4, C22×D5, C4○D20, C2×C5⋊D4, C23.23D10
►On 80 points
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 10 63 72)(2 71 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 20 73 62)(12 61 74 19)(13 18 75 80)(14 79 76 17)(15 16 77 78)(21 24 46 49)(22 48 47 23)(25 40 50 45)(26 44 51 39)(27 38 52 43)(28 42 53 37)(29 36 54 41)(30 60 55 35)(31 34 56 59)(32 58 57 33)
G:=sub<Sym(80)| (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,10,63,72)(2,71,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,20,73,62)(12,61,74,19)(13,18,75,80)(14,79,76,17)(15,16,77,78)(21,24,46,49)(22,48,47,23)(25,40,50,45)(26,44,51,39)(27,38,52,43)(28,42,53,37)(29,36,54,41)(30,60,55,35)(31,34,56,59)(32,58,57,33)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,10,63,72)(2,71,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,20,73,62)(12,61,74,19)(13,18,75,80)(14,79,76,17)(15,16,77,78)(21,24,46,49)(22,48,47,23)(25,40,50,45)(26,44,51,39)(27,38,52,43)(28,42,53,37)(29,36,54,41)(30,60,55,35)(31,34,56,59)(32,58,57,33) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,61),(57,62),(58,63),(59,64),(60,65)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,56),(22,57),(23,58),(24,59),(25,60),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,10,63,72),(2,71,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,20,73,62),(12,61,74,19),(13,18,75,80),(14,79,76,17),(15,16,77,78),(21,24,46,49),(22,48,47,23),(25,40,50,45),(26,44,51,39),(27,38,52,43),(28,42,53,37),(29,36,54,41),(30,60,55,35),(31,34,56,59),(32,58,57,33)]])
C23.23D10 is a maximal subgroup of
(C22×C4)⋊F5 C22⋊C4⋊D10 C42.277D10 C24.27D10 C24.31D10 C10.2- 1+4 C10.52- 1+4 C10.62- 1+4 C42⋊10D10 C42.96D10 C42.104D10 C42⋊16D10 C42.113D10 C42.114D10 C42⋊17D10 C42.115D10 C42.116D10 C42.118D10 C10.422+ 1+4 C10.442+ 1+4 C10.482+ 1+4 C10.742- 1+4 C10.202- 1+4 C10.222- 1+4 C10.582+ 1+4 C10.262- 1+4 C10.792- 1+4 C4⋊C4.197D10 D5×C22.D4 C10.1202+ 1+4 C4⋊C4⋊28D10 C10.852- 1+4 C24.72D10 C24⋊8D10 C10.442- 1+4 C10.1042- 1+4 C10.1452+ 1+4 D6⋊Dic5⋊C2 D10⋊C4⋊S3 (C2×C30).D4 C10.(C2×D12) C23.28D30
C23.23D10 is a maximal quotient of
C10.92(C4×D4) (C2×C42)⋊D5 C24.9D10 C24.14D10 (C2×C10).40D8 C4⋊C4.228D10 C4⋊C4.230D10 C4⋊C4.231D10 (C2×C20).287D4 (C2×C20).288D4 (C2×C20).289D4 (C2×C20).290D4 C4⋊C4.233D10 C4⋊C4.236D10 C24.62D10 C24.63D10 C24.65D10 D6⋊Dic5⋊C2 D10⋊C4⋊S3 (C2×C30).D4 C10.(C2×D12) C23.28D30
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C4○D20 |
| kernel | C23.23D10 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C22×C20 | C2×C10 | C22×C4 | C10 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 8 | 16 |
Matrix representation of C23.23D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 35 | 26 | 0 | 0 |
| 38 | 20 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 22 | 9 |
| 21 | 26 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 9 | 22 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,24,40,0,0,1,17],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[35,38,0,0,26,20,0,0,0,0,19,22,0,0,19,9],[21,24,0,0,26,20,0,0,0,0,19,9,0,0,19,22] >;
C23.23D10 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{10} % in TeX
G:=Group("C2^3.23D10"); // GroupNames label
G:=SmallGroup(160,150);
// by ID
G=gap.SmallGroup(160,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,86,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^9>;
// generators/relations